A Godunov-type scheme has been implemented for the Euler equations governing atmospheric flows. The Riemann problem in Godunov's method is solved with an approximate HLLC Riemann solver. The HLLC (Harten-Lax-van Leer-Contact) Riemann solver is an extension of the HLL (Harten-Lax-van Leer) Riemann solver by Toro. Second-order accuracy in space is achieved by a MUSCL-type (Monotone Upstream Centered Schemes for Conservation Laws) gradient reconstruction after van Leer and the scheme is made total variation diminishing (TVD) with the help of slope limiters. Higher-order accuracy in time is maintained by implementing a two stage explicit Runge-Kutta time-marching scheme. The numerical scheme is evaluated against exact solutions of different benchmark cases in one dimension and idealized test cases in two dimensions. Some benchmark cases for the scalar transport equation are also discussed to demonstrate the accuracy and stability of the scheme.